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Register a new userRegionally proximal relation of order $d$ along arithmetic progressions and nilsystems
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The regionally proximal relation of order $d$ along arithmetic progressions, namely ${bf AP}^{[d]}$ for $din N$, is introduced and investigated. It turns out that if $(X,T)$ is a topological dynamical system with ${bf AP}^{[d]}=Delta$, then each ergodic measure of $(X,T)$ is isomorphic to a $d$-step pro-nilsystem, and thus $(X,T)$ has zero entropy. Moreover, it is shown that if $(X,T)$ is a strictly ergodic distal system with the property that the maximal topological and measurable $d$-step pro-nilsystems are isomorphic, then ${bf AP}^{[d]}={bf RP}^{[d]}$ for each $din {mathbb N}$. It follows that for a minimal $infty$-pro-nilsystem, ${bf AP}^{[d]}={bf RP}^{[d]}$ for each $din {mathbb N}$. An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.
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Let $pi: (X,T)rightarrow (Y,T)$ be a factor map of topological dynamics and $din {mathbb {N}}$. $(Y,T)$ is said to be a $d$-step topological characteristic factor if there exists a dense $G_delta$ set $X_0$ of $X$ such that for each $xin X_0$ the orbit closure $overline{mathcal O}((x, ldots,x), Ttimes T^2times ldots times T^d)$ is $pitimes ldots times pi$ ($d$ times) saturated. In 1994 Eli Glasner studied the topological characteristic factor for minimal systems. For example, it is shown that for a distal minimal system, its largest distal factor of order $d-1$ is its $d$-step topological characteristic factor. In this paper, we generalize Glasners work to the product system of finitely many minimal systems and give its relative version. To prove these results, we need to deal with $(X,T^m)$ for $min {mathbb {N}}$. We will study the structure theorem of $(X,T^m)$. We show that though for a minimal system $(X,T)$ and $min {mathbb {N}}$, $(X,T^m)$ may not be minimal, but we still can have PI-tower for $(X,T^m)$ and in fact it looks the same as the PI tower of $(X,T)$. We give some applications of the results developed. For example, we show that if a minimal system has no nontrivial independent pair along arithmetic progressions of order $d$, then up to a canonically defined proximal extension, it is PI of order $d$; if a minimal system $(X,T)$ has a nontrivial $d$-step topological characteristic factor, then there exist ``many $Delta$-transitive sets of order $d$.
In this paper, we investigate the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions. For positive integers $n$ and $k$, the expression $aw([n],k)$ denotes the smallest number of colors with which the integers ${1,ldots,n}$ can be colored and still guarantee there is a rainbow arithmetic progression of length $k$. We establish that $aw([n],3)=Theta(log n)$ and $aw([n],k)=n^{1-o(1)}$ for $kgeq 4$. For positive integers $n$ and $k$, the expression $aw(Z_n,k)$ denotes the smallest number of colors with which elements of the cyclic group of order $n$ can be colored and still guarantee there is a rainbow arithmetic progression of length $k$. In this setting, arithmetic progressions can wrap around, and $aw(Z_n,3)$ behaves quite differently from $aw([n],3)$, depending on the divisibility of $n$. As shown in [Jungic et al., textit{Combin. Probab. Comput.}, 2003], $aw(Z_{2^m},3) = 3$ for any positive integer $m$. We establish that $aw(Z_n,3)$ can be computed from knowledge of $aw(Z_p,3)$ for all of the prime factors $p$ of $n$. However, for $kgeq 4$, the behavior is similar to the previous case, that is, $aw(Z_n,k)=n^{1-o(1)}$.
For a topological dynamical system $(X, T)$, $linmathbb{N}$ and $xin X$, let $N_l(X)$ and $L_x^l(X)$ be the orbit closures of the diagonal point $(x,x,ldots,x)$ ($l $ times) under the actions $mathcal{G}_{l}$ and $tau_l $ respectively, where $mathcal{G}_{l}$ is generated by $Ttimes Ttimes ldots times T$ ($l $ times) and $tau_l=Ttimes T^2times ldots times T^l$. In this paper, we show that for a minimal system $(X,T)$ and $lin mathbb{N}$, the maximal $d$-step pro-nilfactor of $(N_l(X),mathcal{G}_{l})$ is $(N_l(X_d),mathcal{G}_{l})$, where $pi_d:Xto X/mathbf{RP}^{[d]}=X_d,din mathbb{N}$ is the factor map and $mathbf{RP}^{[d]}$ is the regionally proximal relation of order $d$. Meanwhile, when $(X,T)$ is a minimal nilsystem, we also calculate the pro-nilfactors of $(L_x^l(X),tau_l)$ for almost every $x$ w.r.t. the Haar measure. In particular, there exists a minimal $2$-step nilsystem $(Y,T)$ and a countable set $Omegasubset Y$ such that for $yin Ybackslash Omega$ the maximal equicontinuous factor of $(L_y^2(Y),tau_2)$ is not $(L_{pi_1(y)}^2(Y_{1}),tau_2)$.
We prove that the maximal infinite step pro-nilfactor $X_infty$ of a minimal dynamical system $(X,T)$ is the topological characteristic factor in a certain sense. Namely, we show that by an almost one to one modification of $pi:X rightarrow X_infty$, the induced open extension $pi^*:X^* rightarrow X^*_infty$ has the following property: for $x$ in a dense $G_delta$ set of $X^*$, the orbit closure $L_x=overline{{mathcal{O}}}((x,x,ldots,x), Ttimes T^2times ldots times T^d)$ is $(pi^*)^{(d)}$-saturated, i.e. $L_x=((pi^*)^{(d)})^{-1}(pi^*)^{(d)}(L_x)$. Using results derived from the above fact, we are able to answer several open questions: (1) if $(X,T^k)$ is minimal for some $kge 2$, then for any $din {mathbb N}$ and any $0le j<k$ there is a sequence ${n_i}$ of $mathbb Z$ with $n_iequiv j (text{mod} k)$ such that $T^{n_i}xrightarrow x, T^{2n_i}xrightarrow x, ldots, T^{dn_i}xrightarrow x$ for $x$ in a dense $G_delta$ subset of $X$; (2) if $(X,T)$ is totally minimal, then ${T^{n^2}x:nin {mathbb Z}}$ is dense in $X$ for $x$ in a dense $G_delta$ subset of $X$; (3) for any $dinmathbb N$ and any minimal system, which is an open extension of its maximal distal factor, ${bf RP}^{[d]}={bf AP}^{[d]}$, where the latter is the regionally proximal relation of order $d$ along arithmetic progressions.
Celebrated theorems of Roth and of Matouv{s}ek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $Theta(n^{1/4})$. We study the analogous problem in the $mathbb{Z}_n$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $mathbb{Z}_n$ for all positive integer $n$. We further determine up to a constant factor the discrepancy of arithmetic progressions in $mathbb{Z}_n$ for many $n$. For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $mathbb{Z}_n$ is $Theta(n^{1/3+r_k/(6k)})$, where $r_k in {0,1,2}$ is the remainder when $k$ is divided by $3$. This solves a problem of Hebbinghaus and Srivastav.
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